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Holonomy rigidity for Ricci-flat metrics

  • Bernd Ammann
  • Klaus Kröncke
  • Hartmut Weiss
  • Frederik Witt
Article
  • 57 Downloads

Abstract

On a closed connected oriented manifold M we study the space \(\mathcal {M}_\Vert (M)\) of all Riemannian metrics which admit a non-zero parallel spinor on the universal covering. Such metrics are Ricci-flat, and all known Ricci-flat metrics are of this form. We show the following: The space \(\mathcal {M}_\Vert (M)\) is a smooth submanifold of the space of all metrics and its premoduli space is a smooth finite-dimensional manifold. The holonomy group is locally constant on \(\mathcal {M}_\Vert (M)\). If M is spin, then the dimension of the space of parallel spinors is a locally constant function on \(\mathcal {M}_\Vert (M)\).

Notes

Acknowledgements

We thank X. Dai for discussions about the history of the subject and H.-J. Hein for some discussions related to Tian-Todorov theory.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Bernd Ammann
    • 1
  • Klaus Kröncke
    • 2
  • Hartmut Weiss
    • 3
  • Frederik Witt
    • 4
  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Fachbereich MathematikUniversität HamburgHamburgGermany
  3. 3.Mathematisches SeminarUniversität KielKielGermany
  4. 4.Institut für Geometrie und TopologieUniversität StuttgartStuttgartGermany

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